If the objective function $Z=px+qy,p>0,q>0$ of a linear programming problem attains its optimal value at the points (4, 7) and (5, 5) and $pq = 50$ then
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → $p = 10,q=5$
Optimal value at both points means:
$4p + 7q = 5p + 5q$
$\Rightarrow 7q - 5q = 5p - 4p$
$\Rightarrow 2q = p$
Given: $pq = 50$
$\Rightarrow (2q)\,q = 50$
$\Rightarrow 2q^{2} = 50$
$\Rightarrow q^{2} = 25$
$\Rightarrow q = 5$ (since $q>0$)
$p = 2q = 10$
The values of $p$ and $q$ are $p=10,\; q=5$.